Today

  • Populations and Samples

  • Crime Data

  • Measurement

  • Distributions

  • Conditional Distributions

Populations and samples

 

Terminology

Population: All members of some given set of interest

  • Our goal is usually to learn something about this group

Sample: A subset of the population selected for study

Sampling: Process of selecting members of population into the sample

Sampling frame: Members of population eligible to be in your sample

Observation: One of the sampled units

Variable: A measure describing the observation(s)

 

Your data will contain only observations and variables, but their value is determined by everything else

Example: BCS

To estimate the level of crime victimisation in the United Kingdom, the British Crime Survey surveys a random sample of 3,000 UK household residents per month and asks them about incidents of victimisation (e.g., burglary) in the past year.

Population: Everyone in the United Kingdom

Sample: 3000 household residents

Sampling: Random selection from all UK households

Sampling frame: People living in a known household

Observation: A person

Variable: Past year burglary victimisation

What would these data be bad for?

Types of statistics

Descriptive

  • Describing a sample
  • Summarizing variables

Inferential

  • Learning about the population
  • Testing hypotheses

To make inferences about populations using samples, your sample needs to be representative of your population—at least in the ways that matter

Samples are representative when they have a similar composition to the population along the characteristics you are interested in

 

Next week we’ll be talking about inferential statistics

Crime data

 

Police data

An example: data.police.uk

  • All police-recorded crime in England, Wales, and Northern Ireland
    • A population!
    • Tens/hundreds of thousands of observations
    • Geolocation
    • Small area estimates possible

Advantages:

  • Captures wide array of offenses
  • Often easily available
  • Often all that is available

Challenges:

  • Misses unreported crime
  • Reporting varies widely
  • Quality varies widely

The Dark Figure of Crime

Not all crimes are discovered

  • Victimless crimes
  • Missing person homicides

Not all crimes are reported

  • Varies by place, individuals, crime type

Not all reported are recorded by police

  • Different rules (e.g., hierarchy)
  • Differential treatment
  • Political incentives

An example

Observed Crime Rate = Actual Rate ∗ Probability of Reporting

According to the U.C.R., the incidence of rape nearly doubled from 1973 to 1990. The N.C.V.S., by contrast, shows that it declined by around forty per cent during the same period. Researchers… found that the upward trend in the U.C.R. data correlated with upticks in the number of female police officers, and with the advent of rape crisis centers and reformed investigative styles. It could be, in short, that a modernized approach to the policing of rape drastically increased the frequency with which it was reported while reducing its incidence.

Matthew Hutson. 2020. “The Trouble with Crime Statistics.” The New Yorker

Note the complex process generating the data!

Victimization data

An Example: British Crime Survey

  • Self-reported victimisation
  • Sample of 3,000 per month
    • Great for nationwide estimates
    • Bad for small area estimates

Advantages:

  • Captures unreported crime
  • Can get at exposure and effects
    • How many times
    • How badly injured

Challenges:

  • Reluctance to answer
  • Hidden populations
  • Expense
  • Doesn’t capture victimless crime

Self-reported offense data

An Example: PADS+

  • 716 youth in/around Peterborough
    • Representative sample at age 12
  • Repeated follow-up—with incredible retention
  • Detailed information on life and context

Advantages:

  • Captures unreported crime
  • Captures crime without victim
  • Can get at motivation

Challenges:

  • Reluctance to answer
  • Hidden populations
  • Many crimes very rare
  • Expense

Ideal criminological data

Ideal data would cover every potential crime and what occurred as a result

That is, for a given opportunity, was the crime…

  • Attempted?
  • Successful?
  • Discovered?
  • Reported to police?

And was the offender…

  • Seen or found?
  • Informally sanctioned?
  • Arrested?
  • Incarcerated?

These are all important but nearly never all observed, especially not for a large number of incidents

Types of studies

  • Case study: One unit observed once (perhaps over a period)

  • Cross-section: Many units observed at one time point

  • Time series: One unit observed at multiple times

  • Time-series cross-section:

    • Many units observed at different times
    • Not the same units at each time

  • Panel study:

    • Many units observed at different times
    • Same units observed at each time

 

Each is useful for answering different questions

Measurement

 

What is measurement?

 

The process of connecting an abstract concept to empirical data

 

Research questions must be testable with empirical data

 

Testing thus relies heavily on measurement

 

No research design can overcome bad measures

Types of measures

‍Observables: Objective, externally measurable variables

  • Individual height, occupation, stated beliefs, arrest history
  • Neighborhood population, geographical area, crime rate

Observables can be directly seen and/or experienced

Unobservables: Subjective, internal, or only indirectly measurable

  • Individual self-control, wellness, social class, actual beliefs
  • Neighborhood social capital, disadvantage, collective efficacy

Unobservables do not exist in the real world: they are conceptual

In Criminology, we’re very often interested in unobservables.

Measurement principles

‍Conceptualization: What is X?

  • Define what it constitutes and how it relates to other concepts
  • Goal: Minimize ambiguity

‍Operationalization: How do I measure X?

  • Define observable(s)—indicators—that are related to the concept
  • If single indicator, it is a proxy for the concept
    • e.g. Years of School is a proxy for Education
  • If multiple indicators, they can be combined into composite measure

When reading articles, pay attention to how things are measured!

Indicators and causes

Reflective: The value of the underlying unobservable variable is reflected in the indicators

Formative: The underlying unobservable variable is formed by the combination of the indicators

 

These impact how we estimate measures—a topic for another time!

Let’s see some examples

Individual reflective:

Self-control

Concept: The ability to delay gratification, tolerate frustration, and carefully consider before acting.

Indicators: Would you agree, neither agree nor disagree, or disagree that you…

  • “get upset when you have to wait for something?”
  • “act without stopping to think?”
  • “like to do daring things?”
  • “are impatient—want to have things right away?”
  • “are careful about what you do?”

Assumption: Shared variation in the indicators represents self-control.

Individual formative

Offending frequency scale

Concept: A summary measure of frequency and severity of offending.

Indicators: In the past year, how many times have you…

  • “taken something that wasn’t yours?”
  • “used force to take something from someone else?”
  • “punched, kicked, or pushed someone else?”
  • “attacked someone with a weapon like a knife or gun?”

Assumption: This combination of different types of offending counts provides a useful measure of something.

Neighborhood reflective

Child-centered informal social control

Concept: The shared neighborhood norms and expectations for intervening against child misbehavior.

Indicators: How likely is it that people in your neighborhood would stop it if…

  • “a group of neighborhood children were skipping school and hanging out on a street corner.”
  • “some children were spray-painting graffiti on a local building.”
  • “children were fighting out in the street.”
  • “a child was showing disrespect to an adult.”

Assumption: Shared variation in the measures represents underlying expectations for social control of children.

Neighborhood formative

Concentrated disadvantage

Concept: The spatial clustering of socioeconomic disadvantage.

‍Indicators: Census tract percentage of…

  • “families below the poverty line.”
  • “individuals on public assistance.”
  • “individuals unemployed.”
  • “female-headed households.”
  • “individuals under age 18.”

Assumption: The combination of these measures provides a useful index of disadvantage.

Estimation is hard!


Types of quantitative data

 

Continuous data

Continuous data take ordered, evenly-spaced, numeric values:

  • The difference between 1cm and 2cm is the same as 5cm and 6cm
  • The difference between 20C and 25C is the same as 25C and 30C

If they have a true non-arbitrary zero, they are ratio data

  • If something is 0cm long, it has no length (ratio!)
  • If the temperature is 0C, there isn’t no temperature (not ratio!)
    • The 0 is arbitrary: In Fahrenheit it is 32 degrees!
  • Multiplication and division only work with ratio data
    • 15cm is half as long as 30cm
    • 15C is not half as hot as 30C
      • 15C = 59F = 288K; 30C = 86F = 303K

Discrete or count data

Counts are a special type of numeric data:

  • True zero like ratio data
  • Strictly non-negative
  • Only whole (integer) values

Events or incidents are counts

  • You can’t have negative crimes or half a crime.

We get rates by dividing counts by exposures like population, time, or area

  • Exposures measure what is “at risk” for an event
  • e.g., Homicides per 100,000 people

Ordinal data

Ordinal data can be ranked (ordered), but have no units or even intervals

A common Likert scale item:

  • Strongly Agree
  • Agree
  • Neither Agree nor Disagree
  • Disagree
  • Strongly Disagree

No clear intervals:

The difference between “strongly agree” and “disagree” and between “disagree” and “neither agree nor disagree” may not be the same

This means you cannot add/subtract or multiply/divide ordinal data

You also can’t easily compare one person’s ordinal value to another’s

  • There’s no objective reference point

Nominal (categorical) data

Nominal or categorical data can’t even be ranked. Every value is qualitatively, rather than quantitatively, different:

  • Gender, nationality, occupation

Sometimes we use nominal data to create other types:

  • Occupation into ranked socio-economic status or prestige
  • Nationality into high/middle/low income country

Binary or dichotomous data are a special case of nominal or ordinal data when there are only two categories

  • e.g., alive or dead
  • A useful mean can be calculated for binary data!

Describing data

 

Central tendency

Measures of central tendency are ways to summarize a variable mathematically using a (usually) single number

Mean

  • The “average” or “expected value”
  • Defined only for continuous data

 

Mode

  • The most common single value
  • Defined for all data (even continuous!)

Median

  • The “middle” value or 50th percentile
  • Defined for continuous or ordinal data

Dispersion

Measures of dispersion summarize how spread out values are in a variable

We typically only define these for continuous data

Standard Deviation

  • Roughly the average difference between data values and the mean
    • Low values indicate data are clustered around the mean
    • High values indicate data are spread away from the mean

Variance

  • The square of the standard deviation
  • Not typically interpretable
  • Mainly useful in calculations

Probability distributions

Mathematical functions that give the probability of observing each value

  • This is just how likely it is for a variable to take any given value
  • Common notation: \(P(X)\) (probability of \(X\))

Empirical distributions or sample distributions

  • The set of all observed values of a variable

Theoretical distributions

  • Imagined or expected frequencies based on mathematical parameters
  • We often compare empirical distributions to theoretical ones

We’ll focus on empirical ones today and talk about theoretical ones next week

Categorical

Because they take a limited number of values, we typically summarize categorical data by the count or proportion of each value

Type of Crime Frequency Proportion
Assault 89 0.36
Robbery 24 0.10
Theft 137 0.55

Bar plots are a good way to visualize categorical distributions

Ordinal

Ordinals can be summarized similarly to categoricals, but they have a median—where the cumulative proportion (percentile) passes 0.50.

Fear of Crime Frequency Proportion Cumulative Proportion
Very Low 25 0.08 0.08
Low 100 0.32 0.40
Moderate 75 0.24 0.64
High 60 0.19 0.83
Very High 50 0.16 0.99

Continuous

Continuous data can take a limitless number of values, so we summarize them with measures of central tendency and dispersion.

Mean Median Mode Std. Dev. N
131.14 78.68 33.57 153.32 400

 

We visualize continuous distributions with histograms and density plots

Visualise your data

These all have the same mean, median, and standard deviation!

Distribution Mean Median Std. Dev
Normal 1.0 1.0 1.0
Poisson 1.0 1.0 1.0
Uniform 1.0 1.0 1.0

Quantitative relationships

 

Types of relationships

Association: X and Y tend to “go together”

  • e.g., ice cream sales and violent crime rise at same time
  • May imply common causes, e.g. temperature
  • Does not imply banning ice cream will reduce crime or vice versa
  • Purely observational—makes no assumptions

Effect: X causes Y

  • e.g., clearing a vacant lot reduces nearby violence
  • Implies direction and cause:
    • Clearing more lots will reduce violence there
    • Reducing violence will not clear lots
  • Requires strong assumptions to identify

For now, we’ll focus on associations

Distributions

Associations are really about conditional probabilities

  • Probability \(Y\) takes a particular value when \(X\) takes a particular value
    • Notation: \(P(Y|X)\) is “Probability of Y given X”
  • e.g., if a suspect is white, their probability of arrest is 0.3.
    • \(P(arrest|white) = 0.3\)

A conditional distribution is the distribution a variable takes when a different variable takes some particular value

A joint distribution is the conditional distribution of each variable at every value of the other variable

These are hard to think about but intuitive in action

Cross-tabs

The simplest joint distributions are those between categorical variables, usually displayed as cross-tabulations

Sex Non-Violent Violent
Female 13 7
Male 22 28
Sex Non-Violent Violent
Female 0.65 0.35
Male 0.44 0.56

We can also visualise them with a bar plot

Categorical x continuous

If our association is between a categorical and a continuous variable, we can calculate statistics within each category

Area Mean Crime Rate Std. Dev. N
Rural 5.40 3.78 131
Urban 23.72 12.98 119

Or draw a histogram or density for each category

Continuous x continuous

When both variables are continuous, we usually measure association with a correlation and visualise with a scatterplot

Crime Rate
Pop. Density 0.79

Here I’ve added a regression line to illustrate the correlation

Going Further

We can go pretty far just visualising results!

Soon we’ll get into statistical models to measure and test relationships

But they’re pretty much all just based on cross-tabs, group means, or correlations

What does this show?

 

Correlation cautions

 


He will kill again

Anscombe’s Quartet

Visualisation is important—these all have the same correlation (0.82)!

Dinosaur deceit

So do these! Their \(X\) and \(Y\) values also have the same means and standard deviations!

Wrap-up

  • Takeaways

    • Measurement is complicated but important
    • Quantitative analysis is mainly about distributions
    • Always visualise your data!

  • We’ll pick up with more on relationships next week, and use that to lead into statistical inference